The design of this body is based on inscribed cubes into a dodecahedron. Such cube is used to prove the existence of the didecahedron: one builds tent shaped roofs on each face of a cube. Choosing the dimensions of the roof appropriately, one gets a dodecahedron.
One can find five inscribed cubes in a dodecahedron having 30 faces all together. All of this 30 squares are replaced by wrench shaped faces as seen on the left. The "wrench" is aligned symmetrically along one of the diagonals spreading from vertex to vertex. The other two sharp ends of the wrench protrude so that five of those protrusions (one from each cube) meet in a single point.
The body has the same symmetries as the dodacehedron. The five-fold symmetry is inherited from rotation around the center of the pentagonal faces of the dodcahedron, while the three-fold symmetries are rotations around lines connecting a vertex to the center. (The same as the main diagonal of the inscribed cube.) Both symmetries are visible on the pictures below.
All faces are interleaved, and barely avoid each other. I've also made a model from plywood sheets, but I don't know whether the body could put together if the faces are absolutely rigid.
To build the body, cut 30 of the patterns found in the supplied pdf file. The arrangement is not trivial, so use light glue first, and reinforce the joints after all pieces are in place.
|Five-fold symmetry||Three-fold symmetry|
|Peeking inside||And a nice side view|
and on Yammer CEU.